In the discrete setting, the picture is less complete. To the best of our knowledge, a singular theory in the sense of (I) does not yet exist for maps. This is perhaps because of difficulties relating to the fact that there is no direct analogue for the time rescaling \(\tau = \varepsilon t\)  which leads to an equivalent `slow formulation' of the map (REF ). Thus it is not immediately clear how a to obtain a `reduced map' which describes slow iteration on or close to \(S\) . On the other hand, the perturbation and invariant manifold theory required by (II) for the (discrete analogue of the) normally hyperbolic regime is in principle quite established, dating back at least to the work of Hirsch, Pugh & Shub [1]}, and results on the existence of invariant manifolds and the foliation of the adjacent space in 2-dimensional maps arising in the analysis of fast-slow ODE systems in particular are given in [2]}. We also mention Pötzsche [3]}, Nipp & Stoffer [4]}, [5]} and Shub [6]} (many more references can be found in the books [5]}, [6]}). In many practical situations, however, a direct application of the pre-existing results of the results in [1]} can be difficult due to their generality and the relatively abstract formulation of necessary and sufficient conditions of their applicability. On the other hand, concrete and more applicable formulations such as those in Nipp & Stoffer [5]} depend upon the identification of suitable coordinates, and often require the rather extensive use of nonlinear coordinate transformations and cutoff techniques in order to `prepare' the equations. Thus in many situations, there remains a practical barrier to the application of these results. With regard to perturbation results on the dynamics in the non-normally hyperbolic setting we mention [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [5]}, [19]}.
